Crystal shape/habit/morphology is an important characteristic of crystals (both single crystal and powder specimen) which determines several functional properties and hence their applications. The crystal shape information consists of:
1) The number and types of crystal faces exposed
2) The geometrical area of each exposed crystal face
3) The relative orientation of adjacent crystal faces
4) The volume of the crystal and
5) The 3-D shape of the crystal.
After the discovery of X-rays in 1895 by W. C. Roentgen, X-ray crystallography was developed into a remarkably useful methodology finding applications in a wide variety of areas such as minerals/ores/geology, solid state physics and chemistry, materials science, environmental and pharmaceutical industries, to name a few.
The three-dimensional structure of non-amorphous materials, such as minerals, is defined by regular, repeating planes of atoms that form a crystal lattice. When a focused X-ray beam interacts with these planes of atoms, part of the beam is transmitted, part is absorbed by the sample, part is refracted and scattered and part is diffracted. X-rays are diffracted by each mineral differently depending on what atoms make up the crystal lattice and how these atoms are arranged.
There are two major types of X-ray Diffractometers: single crystal diffractometer and powder diffractometer. As the name suggests, the first kind examines a single crystal of size ranging from centimeters down to micro-meters and even nano-meters. Powder diffractometers use powder specimens which consist of a large number of tiny particles which are either small single crystals or agglomerates of them, the size of these particles ranging usually from micro-meters to nano-meters.
Very sophisticated X-Ray diffraction machines are available today to aid in the study of materials. Examples of the output from the X-ray diffractometer are shown in FIG. 1 [J. Phys. Chem. B, 108, 2887, 2004] and FIG. 2 [J. Phys. Chem. B, 108, 6121, 2004]. What is shown is called an X-ray diffraction pattern and it consists of a series of well-defined peaks at different 2θ-values, where θ is the angle of incidence of the X-ray. The positions of these peaks are described by the Bragg formula [C. Hammond, The Basics of Crystallography and Diffraction, Oxford Science, New York, 2001].2d sin θ=nλ
where
λ=wave-length of X-ray used (Â)
n=order of the diffraction
θ=angle of incidence of the X-ray (degrees)
d=inter-planar spacing along a given crystal direction (Â).
In addition to the position of the diffraction peaks on the 2-axis, the intensity of the diffraction given by the height of the peak and the peak broadening as measured by the width (B) of the peak at half the peak-height are important variables used in the analysis and interpretation of the X-ray diffraction data. In particular, the Scherrer formula [B. D. Cullity, Elements of X-ray Diffraction, Addison-Wesley, Massachusetts, 1956]thkl=0.9*λ/(B Cos θhkl)
where
λ=wave-length of X-ray used.
θhkl=angle of incidence of X-ray on the planes with Miller indices (h, k, l).
B=peak-width at half-maximum.
thkl=thickness of crystal perpendicular to (h, k, l) planes.
is worthy of quote in this context.
A wide range of information may be culled out from XRD patterns using a set of well-developed theoretical/mathematical frame-works. A popular and routine application of an XRD pattern is the identification (also known as characterization) of materials [B. D. Cullity, Elements of X-ray Diffraction, Addison-Wesley, Massachusetts, 1956];
Such an application merely involves a comparison of the XRD pattern of the substance of interest with a library of the XRD patterns of standard reference substances and matching the patterns. This library goes under the name of ICDD (or formerly JCPDS) [www.iucr.ac.uk] data files. More involved analyses of the XRD data will consist in the use of mathematical algorithms and softwares. For example, the internal structure of crystals, consisting of the unit cell parameters (a, b, c and α, β, γ), the atomic/ionic positions within the unit cell and the space-group symmetries, is normally elucidated using fairly advanced mathematical procedures such as the Rietveld refinement [www.bgmn.de for more details].
Though the internal crystal structure determination is the single most prominent application of XRD, there are several other applications to which a mention should be made: qualitative & quantitative chemical analyses, phase purity, phase-diagrams, order-disorder transitions and even mechanical stress.
It must be mentioned that the internal crystal structure of crystals is different from the external physical shapes of crystals which usually goes under the names crystal shape/morphology/habits [F. C. Philips, An Introduction to Crystallography, Longmans, Glasgow, 1971].
Crystal shape is the first thing to come to mind when we think of crystallography (morphology and habits are terms related to the crystal shape). There have been several early attempts to relate the internal structure/symmetry of crystals to crystal shapes/habits. Soon it was realized that there was no straight and quantitative relation which applies to all crystals. Further, the external environment during crystallization plays an important role in determining the shape of particular crystals. For example, there are marked differences of habit for crystallization from vapor and from melt. As Gibbs postulated half a century ago, the physical process of crystal growth is one in which energy factors must be important, i.e. the equilibrium shape of a crystal would be one of minimum total surface free energy for a given volume (Wulff s construction [K. Oura et al., Surface Science An Introduction, Spinger-Verlag, New York, 2003; M. Kitayama et. al., J. Am. Ceram. Soc., 85, 611, 2002]). For kinetically controlled crystal growth, the rates of growth of different faces determine the final crystal shape. It is well known that in kinetically controlled crystal growth fast-growing faces grow “out” of the crystal and disappear, leaving the slow-growing faces to determine the crystal shape. In thermodynamically controlled crystal growth, faces with smaller surface energies dominate the crystal shape. Further, the solvent and other additives may bring about their own “medium effects” by promoting or blocking growth on particular crystal faces.
Traditionally the crystal shape information is gathered from optical microscopes and goniometers [F. C. Philips' book cited above], Scanning electron microscopes (SEM) [A. P. Tsai et al, Japanese Journal of Applied Physics, 26, 1505, 1987; see also FIG. 3], Transmission electron microscopes (TEM/HRTEM) [Calvin J. Curtis et. al, Journal of The Electrochemical Society, 151, A590, 2004], scanning tunneling microscopes (STM) [K. Oura et al., Surface Science An Introduction, Spinger-Verlag, New York, 2003] and the likes, depending on the size of crystal particles. While the optical microscopes and goniometers are used for large crystals (from microns up to centimeters), sub-microns are amenable to the more advanced TEM, HRTEM etc. However, nano-crystal shapes are not easily obtainable from any of these.
It is further noted here that a set of crystal drawing methods exists which draw crystal shapes based on data gathered from goniometers [F. C. Philips' book cited above]. However, goniometers are useful only for macro-sized crystals, whereas the present method applies to micro and nano-sized crystals. [www.shapesoftware.com].
Using equipment such as TEM and HRTEM only qualitative information can be gathered from the visual micrographs or photographs taken by these equipments, without further analysis. In addition, assigning each of the crystal faces to an unique miller index (h, k, l) is not possible. These equipments provide only projected 2D information and not 3D information about the crystal shape. Hence, one cannot view the crystal from arbitrary angles and besides only the crystal faces exposed to the instrument's camera will be accessible for inspection. Equipments like TEM and HRTEM are not only costlier than XRD but also not as common as the latter.